Problem: Let $f$ be a twice differentiable function, and let $f(3)=8$, $f'(3)=0$, and $f''(3)=0$. What occurs in the graph of $f$ at the point $(3,8)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(3,8)$ is a minimum point. (Choice B) B $(3,8)$ is a maximum point. (Choice C) C There's not enough information to tell.
Solution: Since $f'(3)=0$, we know that $x=3$ is a critical point. The second derivative test allows us to analyze what happens in the graph of $f$ at this point according to these three cases: If $f''(3)>0$, the graph of $f$ has a minimum point at $x=3$. If $f''(3)<0$, the graph of $f$ has a maximum point at $x=3$. If $f''(3)=0$, the test is inconclusive. [Why is this so?] We are given that $f''(3)=0$. The test is inconclusive. There's not enough information to tell whether $(3,8)$ is a minimum point, a maximum point, or neither.